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      Questions tagged [stochastic-processes]

      A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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      81 views

      Brownian motion and Durret book

      I have a problem to understand the following simple definition in Durrett book: Brownian motion and martingales in analysis. What does the following mean: $T = \inf \{t: B_t \in A\}$. It seems to ...
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      19 views

      Dependent random variables converging to a density in mean

      Let $X$ be an absolutely continuous r.v. with density $f$ which is continuous on $(0,\infty)$. Consider some a.s. decreasing sequence $Y_n$ bounded by $X$ such that $Y_n\searrow 0$ a.s. I stress that ...
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      55 views

      Existence of stationary stochastic processes with very high correlation

      A question was recently asked by a new user, SomeoneHAHA, and then deleted by the user, after receiving an answer. I think the question and the answer (QA) to it may be of interest to some users. ...
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      Angle between Fleming-Viot type 3-particle system

      Consider $(X^1,X^2,X^3)\in (0,\infty)^3$ with each particle starting at $1$ and moving independently according to Brownian motion until random time $\tau_1:=\min \lbrace t>0: X_{t-}^1\wedge X_{t-}^...
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      1answer
      146 views

      Martingales and intersection of random walks

      Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of independent simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random ...
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      48 views

      Extension of probability space problem: Hilbert space valued process V.S. random field

      Maybe the question should be "Understanding the measurability: Hilbert space valued process V.S. random field" Consider the SPDE $${\rm d}u+\cdots{\rm d}t=\sigma(t,u){\rm d}W.$$ Consider the ...
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      Approximate method to extract behavior of a Laplace transform in an intermediate region

      In the theory of random walks, Tauberian type theorems are often applied to extract the small or large-time behavior from a difficult equation. For example, the Montroll-Weiss formula describing a ...
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      1answer
      185 views

      An application of Itô's formula to an SDE on a Lie group

      I'm trying to understand a calculation in this paper (equation (3.8)). With some details removed, the setup is as follows. Let $G$ be a Lie group, and $g(t)$ a curve in $G$ satisfying the SDE $$dg(t)...
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      1answer
      68 views

      Upper confidence bound for Poisson process rate parameter

      Admittedly, this is an elementary question for mathoverflow. However, I've had no real bites on math and stats.stackexchange so I'm cross-posting. I am interested in computing an upper confidence ...
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      140 views

      Stochastic Calculus vs Stochastic Processes in Finance [closed]

      I'm a second year student, interested in financial mathematics, who's trying to plan out his degree path currently. There's a stochastic processes unit offered in year 3 and a stochastic calculus unit ...
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      Left passage probability of $SLE_8$?

      Schramm's formula on left passage probabilities of $SLE_k$ is stated for $k \in [0,8)$ in theorem 2 here. However, after the statement he remarks that the formula simplifies to $1/2$ for $k = 8$. It ...
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      48 views

      Distances between up and down crosses in Gaussian Processes

      Given a gaussian process $g := \mathcal{GP}\left(\mu, \Sigma \right)$, where $\mu$ is the mean and $\Sigma$ is the covariance function, I am interested in estimating the mean value $L_m$ of the ...
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      45 views

      Lebesgue Integral in SDE

      In the context of proving existence of solutions of S(P)DEs, I've found that few (if any) texts offer significant mention to the deterministic drift term of the form $$ \int_0^t f(s,X(s))ds. $$ If we ...
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      1answer
      103 views

      Bernstein Inequality for continous local martingale

      I'm looking for a simple proof of the following fact, which is somehow Bernstein inequality in continuous time. Let $(M_t)_{t\geq 0}$ be a continuous local martingale. Then : $$P(\sup_{t\in [0,T]}...
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      37 views

      Maximal inequalities of stochastic integral with respect to Poisson random measures

      Let $\Phi$ be a smooth convex function such that $\Psi(x)/|x|\to\infty$ as $|x|\to\infty$, and $\hat N(d z,ds)$ be a compensated Poisson measure on $R^d\times R_+$. Do we have the following inequality:...

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